###### Unsupervised Machine Learning Algorithms

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**Arbitrage pricing theory** (APT) is a theory of asset pricing. It asserts that the expected return of an asset can be expressed as a linear function of multiple systematic risk factors priced by the market. APT was introduced in 1976 by Stephen Ross, and it is based on arbitrage arguments.

APT was developed as an alternative to single-factor pricing models such as the capital asset pricing model (CAPM) and the consumption beta model. These two single-factor pricing models do not explicitly explain the cross-section of expected asset returns.

According to APT, multiple factors can be used to explain the expected rate of return on a risky asset. These include:

- Rate of inflation.
- The growth rate in industrial production.
- Spread between long-term and short-term interest rates.
- Spread between high-grade and low-grade bonds.

- Returns from assets can be explained using systemic factors.
- No arbitrage opportunities exist in well-diversified portfolios. Arbitrage refers to the action of buying an asset in the cheaper market and simultaneously selling that asset in the more expensive market to make a risk-free profit.
- By using diversification, the specific risks can be eliminated from portfolios by the investors.

According to APT, if the three assumptions discussed above hold, then the following equation also holds:

$$ E(R_i)=E(R_Z)+\beta_{i1}\lambda_1+\cdots+\beta_{iK}\lambda_K $$

Where:

\(E(R_i)\) = Expected return on a well-diversified portfolio \(i\).

\(\beta_{ij}\) = Sensitivity for a portfolio i relative to factor \(j\).

\(E(R_Z)\) = Expected rate of return on a portfolio with zero betas (such as risk-free rate of return).

\(\lambda_{j}\) = Risk premium relative to factor \(j\).

## Question

Which of the following statements is the

most accurateabout the APT model?

- Returns from the assets can be explained using systematic factors.
- Only three factors are considered under this model.
- Investors’ portfolio decisions can be made, considering just returns’ means, variances, and correlations.
## Solution

The correct answer is A.The APT assumes that returns from assets can be explained using systematic factors. It also assumes that unsystematic risk can be diversified away in a portfolio and that no arbitrage opportunities exist.

B is incorrect.The factors affecting asset returns are not limited to three. They may include the rate of inflation, the growth rate in industrial production, the spread between long-term and short-term interest rates, and the spread between high-grade and low-grade bonds, among others.

C is incorrect.The CAPM is based on a set of assumptions, including that investors’ portfolio decisions can be made purely based on expected returns (means), variances, and correlations.

Reading 40: Using Multifactor Models

*LOS 40 (a) Describe arbitrage pricing theory (APT), including its underlying assumptions and its relation to multifactor models.*